Force Directed Algorithm

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Force Directed Algorithm

• Adel Alshayji
• 4/28/2005

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What is FD Algorithm?

• Force directed algorithms used to represents
  graphs.
• Force directed algorithms view the graph as a
  virtual physical system.
• The nodes of the graph are bodies of the system.
• These bodies have forces acting on or between
  them.
• These forces are physics-based, and therefore
  have a natural analogy, such as magnetic
  repulsion or gravitational attraction.

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What is for?

• The force directed layout use force directed
  algorithms for drawing large graphs.
• In any graph, the edges can be modeled as
  gravitational attraction and all nodes have an
  electrical repulsion between them.
• It is also possible for the system to simulate
  unnatural forces acting on the bodies, which have
  no direct physical analogy, for example the use
  of a logarithmic distance measure.

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Force Calculation

• Cells are connected by a net of forces.
• The force magnitude between each two nodes is
  proportional to the distance separating them.
• By using Hookes law, forces around each node is
  computed and then equilibrium state is sought.
• New values for the node position is computed and
  the whole net of nodes is rearranged.

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How this Algorithm Work?

• Regardless of the exact nature of the forces in
  the virtual physical system, force directed
  algorithms.
• Compute a locally minimum energy layout of the
  nodes.
• This is usually achieved by computing the forces
  on each node and iterating the system in discrete
  time steps.
• The forces are applied to each node and the
  positions are updated accordingly.

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FD Graph Drawing

• Which layout is nicer?

• An energy model is associated with the graph
  layouts.
• Low energy states correspond to nice layouts.

Energy 1.77x10321
Energy 2.23x106
By Yehuda Koren
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FD Graph Drawing (cont.)
Convergence to global minimum is not guaranteed!

• Graph drawing Energy minimization
• Hence, the drawing algorithm is an iterative
  optimization process

Initial (random) layout
Final (nice) layout
Iteration 1
Iteration 2
Iteration 3
Iteration 4
Iteration 5
Iteration 6
Iteration 7
Iteration 8
Iteration 9
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FD Graph Drawing (cont.)

• This graph has Aesthetical properties
• Proximity preservation
• similar nodes are drawn closely
• Symmetry preservation
• isomorphic sub-graphs
• are drawn identically
• No external influences
• Let the graph speak for itself

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Example 1
A graph drawing through a number of iterations of
a force directed algorithm.
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Example 2

• This series of images show a graph drawing
  from a random layout to the final aesthetically
  pleasant drawing of the graph.

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Scaling with Large Graphs

• Traditional force-directed methods are limited to
  a few hundred nodes
• Problems when drawing large graphs
• Visualization issue not enough drawing area
• Cures dynamic navigation, clustering, fish-eye
  view, hyperbolic space,
• Algorithmic issue convergence to a nice layout
  is too slow
• We concentrate on the algorithmic issue, i.e.,
  the computational complexity (mainly time).

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Complexity

• Complexity per single iteration is O(n2)
• Energy contains at least one term for each node
  pair (repulsive forces)
• Estimated number of iterations to convergence is
  O(n)
• Overall time complexity is O(n3)
• Force directed methods do not scale up well to
  large graphs!

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Limitation

• Force-directed algorithms are often used in graph
  drawing due to their flexibility, ease of
  implementation, and the aesthetically pleasant
  drawings they produce.
• However, classical force directed algorithms are
  unable to handle larger graphs due the inherent N
  squared cost at each time step.
• Where N is the number of bodies in the system.
• The FADE layout paradigm, overcomes this
  computational limitation to allow large graphs to
  be drawn and abstractly represented.

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What is FADE Paradigm?

• In the FADE paradigm, we take the graph
  layout and perform a geometric clustering
  (typically by recursive space decomposition) of
  the locations of the nodes. This process, along
  with implied edge creation, forms a hierarchical
  compound graph.

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How it is Work?

• Compute Initial Layout
• REPEAT
• Compute Edge Forces
• Construct Geometric Clustering
• FOR each node V
• Compute Approximate Non-edga Forces on V
• END FOR
• Move Nodes
• Update Bounding Area/Volume
• UNTIL Stopping Condition
• END

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Performance Improvement

• The performance improvement in the FADE paradigm
  comes from
• Compute forces using recursive decomposition for
  the locations of the nodes, rather than using all
  the nodes directly.
• Different decompositions generate recursive
  geometric clustering of the nodes of the graph.
• The recursive clustering does facilitate a
  dramatic improvement in the performance of force
  directed algorithms.

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Performance Improvement (cont.)

• It allows for multi-level viewing of huge graphs
  at various levels of abstraction.
• As the quality of the drawing improves, the
  quality of clustering, exhibited by the
  decomposition tree, improves to a reasonable
  amount.
• This clustering helps in visual abstraction, when
  it has sufficient quality.

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Spring Embedder
Example of F.D. Method

• Replace edges with springs (zero rest length) ---
  attractive forces
• Replace vertices with electrically charged
  particles, repelling each other --- repulsive
  forces
• Start with a random placement of the vertices,
  then just let the system go until it reaches
  equilibrium.

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Example of Spring Method
Kaufmann and Wagner, 2001
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Animated Example
By, Aaron J. Quigley
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Example in 3D
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Where FD Algorithm Fit?

• Force directed methods and large graphs
• Multi-scale acceleration of force directed
  methods
• Halls graph drawing method(a particular
  force-directed method)
• ACE a multi-scale acceleration of Halls method
• High dimensional embedding a new approach to
  graph drawing

By Yehuda Koren
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Conclusion

• Force-directed algorithms are used in graph
  drawing with limited number of nodes.
• It simplifies graph layout so the graph become
  simple to understand and use.
• It is unable to handle larger graphs due the
  inherent O(N 2) cost at each time step.
• FADE Paradigm, works better with graph drawing
  and improve algorithm performance.
• For higher number of nodes, Multi-scale, Hall,
  High Embedding, and ACE algorithm work better
  than Force Directed Algorithm.

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References

• Dr. Aaron John Quigley, College
  Lecturer,Department of Computer Science,
  University College Dublin, Ireland.
• Yehuda Koren, The Weizmann Institute of Science.
• Andrew Kennings, Electrical and Computer
  Engineering, University of Waterloo.
• Sait, Sadiq M., VLSI physical design automation
  theory and practice.

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Thank You
This animation produced by, Aaron J. Quigley